Let \(\pi _{A}\colon A\rightarrow M\)  be a vector bundle over \(M\) . As in the case of Lie algebroids (as in [1]}), a linear \(E\) -connection on \(A\)  is a map \(\nabla \colon \Gamma (E) \times \Gamma (A) \rightarrow \Gamma (A) \)  that fulfills Koszul conditions, i.e., it is \(\mathcal {F}(M)\) -linear in the first argument and, considering the second argument, it is a 1-derivation according to \(\mathcal {F}(M)\)  (see [2]}). The \(E\) -curvature of \(\nabla \)  is the \(\mathcal {F}(M)\) -linear map \(R\colon \Gamma (E) \times \Gamma (E) \times \Gamma (A) \rightarrow \Gamma (A) \) , \(R(X,Y) B=\nabla _{X}\nabla _{Y}B-\nabla _{Y}\nabla _{X}B-\nabla _{[ X,Y]_{A}}B\) .
Consider now bases of local sections over the same open subsets \(U\subset M\) , \(\lbrace  s_{\alpha }\rbrace  _{\alpha =1,\dots ,k}\subset \Gamma ( E_{U}) \)  and \(\lbrace  \bar{s}_{a}\rbrace  _{a=1,\dots ,n}\subset \Gamma (A_{U})\) . Let us consider the dual bases \(\lbrace  \omega ^{\alpha }\rbrace  _{\alpha =1,\dots ,k}\subset \Gamma ( E_{U}^{\ast }) \)  and \(\lbrace  \bar{\omega }^{a}\rbrace  _{a=1,\dots ,n}\subset \Gamma ( A_{U}^{\ast }) \) . A linear \(E\) -connection on \(A\)  gives the local functions \(\Gamma _{\beta b}^{a}=\bar{\omega }^{a}( \nabla _{s_{\beta }}\bar{s}_{b})\) , \(R_{\alpha \beta b}^{a}=\bar{\omega }^{a}( R(s_{\alpha },s_{\beta })\bar{s}_{b}) \)  and the local forms \(\theta _{b}^{a}=\Gamma _{\beta b}^{a}\omega ^{\beta }\in \Gamma ( E_{U}^{\ast }) \)  and \(R_{b}^{a}=\frac{1}{2}R_{\alpha \beta b}^{a}\omega ^{\alpha }\wedge \omega ^{\beta }\) . The following Cartan formula follows by a straightforward verification.
